A110048 -id:A110048 - cp's OEIS frontend

A097076 Expansion of g.f. x/(1 - x - 3*x^2 - x^3).

0, 1, 1, 4, 8, 21, 49, 120, 288, 697, 1681, 4060, 9800, 23661, 57121, 137904, 332928, 803761, 1940449, 4684660, 11309768, 27304197, 65918161, 159140520, 384199200, 927538921, 2239277041, 5406093004, 13051463048, 31509019101, 76069501249, 183648021600

Offset: 0

Paul Barry, Jul 22 2004

Counts walks of length n between two vertices of a triangle, when a loop has been added at the third vertex.
a(n) is the center term of the 3 X 3 matrix [0,1,0; 0,0,1; 1,3,1]^n. - Gary W. Adamson, May 30 2008
Starting (1, 1, 4, 8, 21, ...) = row sums of triangle A157898. - Gary W. Adamson, Mar 08 2009
Convolution of Pell(n) = A000129(n) and (-1)^n. - Paul Barry, Oct 22 2009
a(n+1) is the number of ways to choose points on a 2 X n lattice eliminating the upper left and lower right corners such that the points are not adjacent to each other. (See A375726 for proof) - Yifan Xie, Aug 25 2024
a(n+1) is the number of compositions (ordered partitions) of n into parts 1, 2, and 3 where there are three kinds of part 2. - Joerg Arndt, Aug 27 2024

G. C. Greubel, Table of n, a(n) for n = 0..1000
J. Bodeen, S. Butler, T. Kim, X. Sun and S. Wang, Tiling a strip with triangles, El. J. Combinat. 21 (1) (2014) P1.7.
Mark Shattuck, Combinatorial Proofs of Some Formulas for Triangular Tilings, Journal of Integer Sequences, 17 (2014), #14.5.5.
Index entries for linear recurrences with constant coefficients, signature (1,3,1).

Cf. A000129, A001333, A051927, A097075, A110048, A157898, A375726.

[(Evaluate(DicksonFirst(n,-1), 2) -2*(-1)^n)/4: n in [0..40]]; // G. C. Greubel, Aug 18 2022

CoefficientList[Series[x/(1-x-3x^2-x^3),{x,0,40}],x] (* or *) LinearRecurrence[{1,3,1},{0,1,1},40]  (* Vladimir Joseph Stephan Orlovsky, Jan 30 2012 *)

[(lucas_number2(n,2,-1) -2*(-1)^n)/4 for n in (0..40)] # G. C. Greubel, Aug 18 2022

a(n) = ( (1+sqrt(2))^n + (1-sqrt(2))^n - 2*(-1)^n )/4.
a(n) = a(n-1) + 3*a(n-2) + a(n-3). [corrected by Paul Curtz, Mar 04 2008]
a(n) = (Sum_{k=0..floor(n/2)} binomial(n, 2*k)*2^k)/2 - (-1)^n/2.
a(n) = (A001333(n) - (-1)^n)/2.
a(n) = Sum_{k=0..n} (-1)^k*Pell(n-k). - Paul Barry, Oct 22 2009
From R. J. Mathar, Jul 06 2011: (Start)
G.f.: x / ( (1+x)*(1-2*x-x^2) ).
a(n) + a(n+1) = A000129(n+1). (End)
E.g.f.: (exp(x)*cosh(sqrt(2)*x) - cosh(x) + sinh(x))/2. - Stefano Spezia, Mar 31 2024

A086346 On a 3 X 3 board, the number of n-move paths for a chess king ending in a given corner square.

Original entry on oeis.org

1, 3, 18, 80, 400, 1904, 9248, 44544, 215296, 1039104, 5018112, 24227840, 116985856, 564850688, 2727354368, 13168803840, 63584665600, 307013812224, 1482394042368, 7157631156224, 34560101318656, 166870928850944, 805724122775552, 3890380202311680, 18784417308737536, 90699190027419648

Offset: 0

Zak Seidov, Jul 17 2003

From Johannes W. Meijer, Aug 01 2010: (Start)
The a(n) represent the number of n-move paths of a chess king on a 3 X 3 board that end or start in a given corner square m (m = 1, 3, 7, 9). To determine the a(n) we can either sum the components of the column vector A^n[k,m], with A the adjacency matrix of the king's graph, or we can sum the components of the row vector A^n[m,k], see the Maple program.
Inverse binomial transform of A079291 (without the leading 0).
(End)
From R. J. Mathar, Oct 12 2010: (Start)
The row n=3 of an array counting king walks on an n X n board with k steps, starting from a corner:
  1, 3,  9,  27,  81,  243,   729,   2187,    6561,    19683,    59049, ...;
  1, 3, 18,  80, 400, 1904,  9248,  44544,  215296,  1039104,  5018112, ...;
  1, 3, 18, 105, 615, 3600, 21075, 123375,  722250,  4228125, 24751875, ...;
  1, 3, 18, 105, 684, 4359, 28278, 182349, 1179792,  7622667, 49283802, ...;
  1, 3, 18, 105, 684, 4550, 30807, 209867, 1434279,  9815190, 67209723, ...;
  1, 3, 18, 105, 684, 4550, 31340, 218056, 1533712, 10829360, 76720288, ...;
  1, 3, 18, 105, 684, 4550, 31340, 219555, 1559835, 11177190, 80573373, ...;
  1, 3, 18, 105, 684, 4550, 31340, 219555, 1564080, 11259785, 81765550, ...;
  1, 3, 18, 105, 684, 4550, 31340, 219555, 1564080, 11271876, 82025163, ...;
  1, 3, 18, 105, 684, 4550, 31340, 219555, 1564080, 11271876, 82059768, ...;
  1, 3, 18, 105, 684, 4550, 31340, 219555, 1564080, 11271876, 82059768, ...;
The partial sums along the rows are documented in A123109 (king walks with between 1 and k steps). (End)

Gary Chartrand, Introductory Graph Theory, pp. 217-221, 1984. [From Johannes W. Meijer, Aug 01 2010]

G. C. Greubel, Table of n, a(n) for n = 0..1000
Mike Oakes, KingMovesForPrimes.
Zak Seidov et al., New puzzle? King moves for primes, digest of 28 messages in primenumbers group, Jul 13 - Jul 23, 2003. [Cached copy]
Zak Seidov, KingMovesForPrimes.
Sleephound, KingMovesForPrimes.
Index entries for linear recurrences with constant coefficients, signature (2,12,8).

Cf. A086347, A086348, A086349.
Cf. A179596. - Johannes W. Meijer, Aug 01 2010
Cf. A002203, A057087, A079291, A084128, A110048, A123109.

[2^(n-3)*(Evaluate(DicksonFirst(n+2,-1), 2) +2*(-1)^n): n in [0..30]]; // G. C. Greubel, Aug 18 2022

with(LinearAlgebra):
nmax:=19; m:=1;
A[5]:= [1, 1, 1, 1, 0, 1, 1, 1, 1]:
A:=Matrix([[0, 1, 0, 1, 1, 0, 0, 0, 0], [1, 0, 1, 1, 1, 1, 0, 0, 0], [0, 1, 0, 0, 1, 1, 0, 0, 0], [1, 1, 0, 0, 1, 0, 1, 1, 0], A[5], [0, 1, 1, 0, 1, 0, 0, 1, 1], [0, 0, 0, 1, 1, 0, 0, 1, 0], [0, 0, 0, 1, 1, 1, 1, 0, 1], [0, 0, 0, 0, 1, 1, 0, 1, 0]]):
for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m, k], k=1..9): od: seq(a(n), n=0..nmax); # Johannes W. Meijer, Aug 01 2010

Table[(1/32)(2(-2)^(n+2)+(2+Sqrt[8])^(n+2)+(2-Sqrt[8])^(n+2)), {n, 0, 19}] // FullSimplify
LinearRecurrence[{2,12,8}, {1,3,18}, 31] (* G. C. Greubel, Aug 18 2022 *)

Vec((1+x)/((1+2*x)*(1-4*x-4*x^2))+O(x^30)) \\ Joerg Arndt, Jan 29 2024

[2^(n-3)*(lucas_number2(n+2,2,-1) +2*(-1)^n) for n in (0..30)] # G. C. Greubel, Aug 18 2022

a(n) = (1/32)*(2*(-2)^(n+2) + (2+sqrt(8))^(n+2) + (2-sqrt(8))^(n+2)).
From R. J. Mathar, Jul 22 2010: (Start)
a(n) = 2*a(n-1) + 12*a(n-2) + 8*a(n-3).
G.f.: (1+x) / ( (1+2*x)*(1-4*x-4*x^2) ).
a(n) = (2*A057087(n-1) + 3*A057087(n) + (-2)^n)/4. (End)
Limit_{k->oo} a(n+k)/a(k) = A084128(n) + 2*A057087(n-1)*sqrt(2). - Johannes W. Meijer, Aug 01 2010
a(n) = A110048(n) + A110048(n-1). - R. J. Mathar, Mar 08 2021
a(n) = 2^(n-3)*(A002203(n+2) + 2*(-1)^n). - G. C. Greubel, Aug 18 2022

Offset changed and edited by Johannes W. Meijer, Jul 15 2010

A110050 Expansion of (2+9x-24x^3+16x^4-30x^2) / ((1-x)(2x+1)(2x-1)(4x^2+4*x-1)).

Original entry on oeis.org

2, 19, 73, 369, 1697, 8241, 39441, 190609, 918929, 4437649, 21421201, 103433361, 499397777, 2411316369, 11642774673, 56216331409, 271436096657, 1310609581201, 6328181400721, 30555163403409, 147533373973649

Offset: 0

Creighton Dement, Jul 10 2005

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,4,-24,0,16).

Cf. A110051, A110048.

seriestolist(series((2+9*x-24*x^3+16*x^4-30*x^2)/((1-x)*(2*x+1)*(2*x-1)*(4*x^2+4*x-1)), x=0,25)); -or- Floretion Algebra Multiplication Program, FAMP Code: -kbasejrokseq[A*B] with A = + 'i - .5'j + .5'k - .5j' + .5k' - 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki' and B = - .5'i + .5'j + 'k - .5i' + .5j' - 'kk' - .5'ik' - .5'jk' - .5'ki' - .5'kj'; RokType: Y[15] = Y[15] + 1/2

Vec((2 + 9*x - 30*x^2 - 24*x^3 + 16*x^4) / ((1 - x)*(1 - 2*x)*(1 + 2*x)*(1 - 4*x - 4*x^2)) + O(x^25)) \\ Colin Barker, May 11 2019

a(n) = 5*a(n-1) + 4*a(n-2) - 24*a(n-3) + 16*a(n-5) for n>4. - Colin Barker, May 11 2019

A110046 Expansion of (1+4x-12x^2-16x^3)/((2x+1)(2x-1)(4x^2+4*x-1)).

Original entry on oeis.org

1, 8, 28, 144, 656, 3200, 15296, 73984, 356608, 1722368, 8313856, 40144896, 193826816, 935886848, 4518821888, 21818834944, 105350496256, 508677324800, 2456110759936, 11859152338944, 57261050298368, 276480810549248

Offset: 0

Creighton Dement, Jul 10 2005

Colin Barker, Table of n, a(n) for n = 0..1000
Robert Munafo, Sequences Related to Floretions
Index entries for linear recurrences with constant coefficients, signature (4, 8, -16, -16).

Cf. A110047, A110048, A110049, A110050.

seriestolist(series((1+4*x-12*x^2-16*x^3)/((2*x+1)*(2*x-1)*(4*x^2+4*x-1)), x=0,25)); -or- Floretion Algebra Multiplication Program, FAMP Code: tesseq[A*B] with A = + 'i - .5'j + .5'k - .5j' + .5k' - 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki' and B = - .5'i + .5'j + 'k - .5i' + .5j' - 'kk' - .5'ik' - .5'jk' - .5'ki' - .5'kj'

CoefficientList[Series[(1+4x-12x^2-16x^3)/((2x+1)(2x-1)(4x^2+4x-1)),{x,0,40}],x] (* or *) LinearRecurrence[{4,8,-16,-16},{1,8,28,144},40] (* Harvey P. Dale, Jun 12 2016 *)

Vec((1 + 4*x - 12*x^2 - 16*x^3) / ((1 - 2*x)*(1 + 2*x)*(1 - 4*x - 4*x^2)) + O(x^40)) \\ Colin Barker, May 01 2019

From Colin Barker, May 01 2019: (Start)
a(n) = ((2 - 2*sqrt(2))^(1+n) + 2*(-(-2)^n + 2^n + 2^n*(1+sqrt(2))^(1+n))) / 4.
a(n) = 4*a(n-1) + 8*a(n-2) - 16*a(n-3) - 16*a(n-4) for n>3.
(End)

A110047 Expansion of (1+2x-4x^2)/((2x+1)(2x-1)(4x^2+4x-1)).

Original entry on oeis.org

1, 6, 28, 144, 688, 3360, 16192, 78336, 378112, 1826304, 8817664, 42577920, 205582336, 992649216, 4792926208, 23142334464, 111741042688, 539533639680, 2605098729472, 12578530000896, 60734514921472, 293252181786624, 1415946786832384, 6836795882864640

Offset: 0

Creighton Dement, Jul 10 2005

Note (see program code): ibaseseq[A*B] = A057087, basejseq[A*B] = A099582, tesseq[A*B] = A110046.

Matthew House, Table of n, a(n) for n = 0..1454
Robert Munafo, Sequences Related to Floretions
Index entries for linear recurrences with constant coefficients, signature (4,8,-16,-16).

Cf. A110046, A110048, A110049.

seriestolist(series((1+2*x-4*x^2)/((2*x+1)*(2*x-1)*(4*x^2+4*x-1)), x=0,25)); -or- Floretion Algebra Multiplication Program, FAMP Code: -kbasekseq[A*B] with A = + 'i - .5'j + .5'k - .5j' + .5k' - 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki' and B = - .5'i + .5'j + 'k - .5i' + .5j' - 'kk' - .5'ik' - .5'jk' - .5'ki' - .5'kj'

CoefficientList[Series[(1 + 2 x - 4 x^2)/((2 x + 1)(2 x - 1)(4 x^2 + 4 x - 1)), {x, 0, 21}], x] (* or *)
LinearRecurrence[{4, 8, -16, -16}, {1, 6, 28, 144}, 22] (* Michael De Vlieger, Feb 17 2017 *)

Vec((1+2*x-4*x^2) / ((2*x+1)*(2*x-1)*(4*x^2+4*x-1)) + O(x^30)) \\ Colin Barker, Feb 17 2017

a(n) = 4*a(n-1) + 8*a(n-2) - 16*a(n-3) - 16*a(n-4). - Matthew House, Feb 17 2017

a(n) = (-3*(2-2*sqrt(2))^n*(-2+sqrt(2)) + 2^n*(-2*(1+(-1)^n)+3*(1+sqrt(2))^n*(2+sqrt(2)))) / 8. - Colin Barker, Feb 17 2017

Definition corrected by Matthew House, Feb 17 2017

cp's OEIS Frontend

A097076 Expansion of g.f. x/(1 - x - 3*x^2 - x^3).

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A086346 On a 3 X 3 board, the number of n-move paths for a chess king ending in a given corner square.

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A110050 Expansion of (2+9*x-24*x^3+16*x^4-30*x^2) / ((1-x)*(2*x+1)*(2*x-1)*(4*x^2+4*x-1)).

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A110046 Expansion of (1+4*x-12*x^2-16*x^3)/((2*x+1)*(2*x-1)*(4*x^2+4*x-1)).

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A110047 Expansion of (1+2*x-4*x^2)/((2*x+1)*(2*x-1)*(4*x^2+4*x-1)).

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A110050 Expansion of (2+9x-24x^3+16x^4-30x^2) / ((1-x)(2x+1)(2x-1)(4x^2+4*x-1)).

A110046 Expansion of (1+4x-12x^2-16x^3)/((2x+1)(2x-1)(4x^2+4*x-1)).

A110047 Expansion of (1+2x-4x^2)/((2x+1)(2x-1)(4x^2+4x-1)).